# nLab noncommutative geometry

## Theorems

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

## Relation to physics

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## Idea

Spaces can be characterized by their algebras of functions (see at Isbell duality - table).

Using this duality between space and quantity one can define generalized spaces in terms of generalizations of their algebras of functions.

The idea of noncommutative geometry is to encode everything about the geometry of a space algebraically and then allow all commutative function algebras to be generalized to possibly non-commutative algebras.

More generally, noncommutative geometry means replacing the space by some structure carried by an entity (or a collection of entities) living on that would-be space. The entity may be for example a function, vector bundle, coherent sheaf, a complex of sheaves and KK-theory class. Objects organize into associative algebras, operator algebras, categories, higher categories ($k$-linear or not) and so on; and sometimes such a collection represents a space. Reconstruction theorems are theorems on construction of a genuine (say topological) “underlying” space of such an entity or collection. Spectral theories are procedures (sometimes functors, often not) which recover some form of underlying space called spectrum, often just partially or under strong assumptions on (the data determining) the noncommutative space.

Under the process of forming groupoid convolution algebras a good bit of commutative but higher geometry translates into noncommutative geometry. This is for instance the origin of the role of noncommutative geometry in twisted K-theory.

## Definitions

### Connes’ noncommutative geometry

A particular and most prominent realization of the program of noncommutative geometry has been lead by Alain Connes. This is really “spectral?” and possibly non-commutative Riemannian geometry, where the metric structure on a possibly noncommutative “algebra of functions” is encoded by a structure called a spectral triple (see there for more).

The central ingredients in Connes’ noncommutative geometry are

• the idea to characterize a (noncommutative) space by a C-star algebra $A$, to be thought of as the $C^*$-algebra of global functions? on that space; this approach has been occasionally considered earlier e.g. in the book of Semadeni on Banach algebras.

• There is a refined, quantized differential calculus where the differential is given by a commutator formula involving a Fredholm operator; the setup in which this is taken place involves cyclic cocycles discovered by Tsygan and Connes.

• the noncommutative analog of the structure of a Riemannian manifold with a spin structure in terms of generalized Dirac operators $D$ acting on a representation space of the algebra $A$. Metric information on the space is then encoded in the spectrum of $D$.

For that reason Connes‘ noncommutative manifolds are well described as spectral geometry. The Dirac operator is, of course, very much related to the quantized differential calculus of Connes.

• Noncommutative measure spaces are represented by noncommutative von Neumann algebras.

In noncommutative geometry various homotopical and (co)homological invariants were introduced by large amount of improvisation, similar to the beginnings of algebraic topology, but more recently there are few systematic approaches to homotopy theory emerging. See model structure on operator algebras.

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

## References

### General

Exposition:

The quantization program and the study of integrable systems brought about a number of interesting examples (in the early phase by G. Kac, Sklyanin, Fadeev, Drinfel’d, Woronowicz, Jimbo, Manin, Reshetikin, Lusztig, Majid and others) with group-like flavour, the study of quantum groups which were studied in a number of formalisms from operator algebraic to algebro-geometric and purely categorical. Actions are very important in noncommutative geometry, and are some of the main examples in Connes’ school like group(oid) $C^*$-algebras, crossed product operator algebras and the study of functions on orbifolds and foliations. They also play an important role in equivariant noncommutative algebraic geometry; cf. the central notions like Hopf–Galois extension and entwining structure.